Trigonometic Ratios (SOH CAH TOA): Missing Side Lengths Worksheet ... - Free Printable

Here are the solutions for the trigonometry problems. Remember to set your calculator to Degrees mode and round your answers to 3 significant figures.

Section A

1)
* Identify sides: We have the hypotenuse (12) and want the opposite side ($x$) relative to the $35^\circ$ angle.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(35^\circ) = \frac{x}{12} $$
$$ x = 12 \times \sin(35^\circ) $$
$$ x \approx 6.8829... $$
* Answer: $x = 6.88$ cm

2)
* Identify sides: We have the hypotenuse (9) and want the adjacent side ($x$) relative to the $67^\circ$ angle.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(67^\circ) = \frac{x}{9} $$
$$ x = 9 \times \cos(67^\circ) $$
$$ x \approx 3.5166... $$
* Answer: $x = 3.52$ cm

3)
* Identify sides: We have the adjacent side (16) and want the opposite side ($x$) relative to the $49^\circ$ angle.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(49^\circ) = \frac{x}{16} $$
$$ x = 16 \times \tan(49^\circ) $$
$$ x \approx 18.405... $$
* Answer: $x = 18.4$ cm

4)
* Identify sides: The angle is at the top left ($62^\circ$). The side labeled 17 is Opposite to this angle. The side labeled $x$ is Adjacent.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(62^\circ) = \frac{17}{x} $$
$$ x = \frac{17}{\tan(62^\circ)} $$
$$ x \approx 9.039... $$
* Answer: $x = 9.04$ cm

5)
* Identify sides: The angle is at the top ($27^\circ$). The side labeled 5 is Adjacent. The side labeled $x$ is Opposite.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(27^\circ) = \frac{x}{5} $$
$$ x = 5 \times \tan(27^\circ) $$
$$ x \approx 2.547... $$
* Answer: $x = 2.55$ cm

6)
* Identify sides: Relative to the $33^\circ$ angle, $x$ is Opposite and 21 is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(33^\circ) = \frac{x}{21} $$
$$ x = 21 \times \sin(33^\circ) $$
$$ x \approx 11.439... $$
* Answer: $x = 11.4$ cm

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Section B

1)
* Identify sides: Relative to the $15^\circ$ angle, 14 is Adjacent and $x$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(15^\circ) = \frac{14}{x} $$
$$ x = \frac{14}{\cos(15^\circ)} $$
$$ x \approx 14.491... $$
* Answer: $x = 14.5$ cm

2)
* Identify sides: Relative to the $71^\circ$ angle, 19 is Opposite and $x$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(71^\circ) = \frac{19}{x} $$
$$ x = \frac{19}{\sin(71^\circ)} $$
$$ x \approx 20.116... $$
* Answer: $x = 20.1$ cm

3)
* Identify sides: Relative to the $42^\circ$ angle, 5 is Opposite and $x$ is Adjacent.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(42^\circ) = \frac{5}{x} $$
$$ x = \frac{5}{\tan(42^\circ)} $$
$$ x \approx 5.553... $$
* Answer: $x = 5.55$ cm

4)
* Identify sides: Relative to the $58^\circ$ angle, 11 is Adjacent and $x$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Calculation:
$$ \cos(58^\circ) = \frac{11}{x} $$
$$ x = \frac{11}{\cos(58^\circ)} $$
$$ x \approx 20.758... $$
* Answer: $x = 20.8$ cm

5)
* Identify sides: Relative to the $20^\circ$ angle, 7 is Opposite and $x$ is Adjacent.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Calculation:
$$ \tan(20^\circ) = \frac{7}{x} $$
$$ x = \frac{7}{\tan(20^\circ)} $$
$$ x \approx 19.230... $$
* Answer: $x = 19.2$ cm

6)
* Identify sides: Relative to the $36^\circ$ angle, 24 is Opposite and $x$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Calculation:
$$ \sin(36^\circ) = \frac{24}{x} $$
$$ x = \frac{24}{\sin(36^\circ)} $$
$$ x \approx 40.831... $$
* Answer: $x = 40.8$ cm

Final Answer:
Section A
1) 6.88
2) 3.52
3) 18.4
4) 9.04
5) 2.55
6) 11.4

Section B
1) 14.5
2) 20.1
3) 5.55
4) 20.8
5) 19.2
6) 40.8